An Improved IP Formulation for the Uncapacitated Facility Location Problem: Capitalizing on Objective Function Structure

Abstract

In this paper we examine the Uncapacitated Facility Location Problem (UFLP) with a special structure of the objective function coefficients. For each customer the set of potential locations can be partitioned into subsets such that the objective function coefficients in each are identical. This structure exists in many applications, including the Maximum Cover Location Problem. For the problems possessing this structure, we develop a new integer programming formulation that has all the desirable properties of the standard formulation, but with substantially smaller dimensionality, leading to significant improvement in computational times. While our formulation applies to any instance of the UFLP, the reduction in dimensionality depends on the degree to which this special structure is present.

A Generalized Model for Locating Facilities on a Network with Flow-Based Demand

“…First, when all decision variables are binary, FILM in general performs better than FILAM, except for p=13. This result fits the statement (Berman and Krass 2005) that when the two models are equivalent and all decision variables are integer, the MCLM-type formulation (FILM) in general performs better than the UFLP-type formulation (GFIM). Second, we investigate the "integer-friendly" property of FILM.…”

“…The literature has reported the computational behaviour of MCLM and UFLP. For example, Berman and Krass (2005) pointed out when the two models are equivalent and all decision variables are integer, an MCLM-type formulation in general performs better than a UFLP-type formulation. Most literature is based on point-based demand data exclusively.…”

The Ordered Gradual Covering Location Problem on a Network

“…The following formulation directly extends the standard UFLP formulation (e.g. see Berman and Krass [2]) and is similar to formulation (OMP 1 ) presented in Nickel and Puerto [13].…”

“…This formulation is based on the one in Berman and Krass [2] as well as Marin, Nickel, Puerto and Velten [11]. The main idea is that in order to compute the contribution of customer node v i ∈ V to the objective function, it is not necessary to know where the customer is covered from -we only need to know the coverage level the customer receives.…”

“…The facility locations are then chosen so as to minimize this total travel distance. The OMP interjects two additional steps into this process: (1A) "sorting", where customer travel distances are sorted from smallest to largest, and (1B) "weighting"where the i-th smallest customer travel distance is multiplied by the weight λ i ; the weighted costs are now added up as in step (2) above. This allows us to represent the standard median objective (by choosing weights λ i = 1 for all i), the center objective (by choosing weights λ i = 0 for the first n − 1 components and λ n = 1 for the last component, we obtain the largest travel distance), and the cent-dian objective (to obtain a convex combination of the median and center objectives with weight α ∈ (0, 1), set λ i = α for all the first n − 1 components and λ n = 1 for the last component).…”